You are right: the middle box will start moving when P > 2µW. However, if P exceeds 2µW just a little, the top box will move along with the middle one; there will be no sliding. Only when P is big enough (note that the problem does not make a upper bound on P), the top box will not be able to keep up with the acceleration of the middle box.

Assuming the two upper boxes move together, the total force on the middle box is P - 2µW, and the acceleration of the upper boxes is (P - 2µW) / 2m, where m = W / g. Compare this with the acceleration that the force of friction can exert on the top box.